Note on Some Convolved Power Sums

Neuman and Schonbach have obtained explicit formulas for the sum \[S(i,j;N) = \sum\limits_{k = 0}^N {k_i (N - k)^i } \quad (i,j \geqq 0)\] by using known results involving Bernoulli numbers. In the present paper the functions \[\begin{gathered} S(i,j;N) = \sum\limits_{k = 0}^N {(k + a)^i (N - k - a)^j } \qquad (i,j \geqq 0), \hfill \\ S'(i,j;N;a) = iS(i - 1,j;N;a) - jS(i,j - 1;N;a), \hfill \\ \end{gathered} \] where a is arbitrary, are evaluated. The evaluation of $S'(i,j;N;a)$ makes use of an appropriate generating function. The final formula for $S(i,j;N;a)$ reduces to the Neuman–Schonbach result when $a = 0$. In addition it is shown that the sum $S(i,j;N)$ is closely related to the Eulerian numbers.